given a language L that we know is in RE and therefore has an enumertor, can we use that enumerator in order to check if a word is in L? or we only know such enumertor exisist, but we don't have it for usage?

If the language L is given to you, you can say that if it is in RE and is infinite - there exists an enumerator f, and use it.
How would you define "to check"? You obviously cannot do this and hope to always halt.

if L is in RE , but finite it doesn't have an enumrator? why?
the slides from the lecture show that a language L is in RE if and only if L has an enumrator, with no regard to wether the language is finite or infinite.